US6108647A - Method, apparatus and programmed medium for approximating the data cube and obtaining approximate answers to queries in relational databases - Google Patents
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- US6108647A US6108647A US09/082,057 US8205798A US6108647A US 6108647 A US6108647 A US 6108647A US 8205798 A US8205798 A US 8205798A US 6108647 A US6108647 A US 6108647A
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F16/00—Information retrieval; Database structures therefor; File system structures therefor
- G06F16/20—Information retrieval; Database structures therefor; File system structures therefor of structured data, e.g. relational data
- G06F16/24—Querying
- G06F16/245—Query processing
- G06F16/2453—Query optimisation
- G06F16/24534—Query rewriting; Transformation
- G06F16/24539—Query rewriting; Transformation using cached or materialised query results
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F16/00—Information retrieval; Database structures therefor; File system structures therefor
- G06F16/20—Information retrieval; Database structures therefor; File system structures therefor of structured data, e.g. relational data
- G06F16/24—Querying
- G06F16/245—Query processing
- G06F16/2458—Special types of queries, e.g. statistical queries, fuzzy queries or distributed queries
- G06F16/2462—Approximate or statistical queries
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
- Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10S707/00—Data processing: database and file management or data structures
- Y10S707/99931—Database or file accessing
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
- Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10S707/00—Data processing: database and file management or data structures
- Y10S707/99931—Database or file accessing
- Y10S707/99932—Access augmentation or optimizing
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
- Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10S707/00—Data processing: database and file management or data structures
- Y10S707/99931—Database or file accessing
- Y10S707/99933—Query processing, i.e. searching
- Y10S707/99936—Pattern matching access
Definitions
- the present invention relates to relational databases and more specifically, to obtaining approximate answers to aggregate queries in relational databases.
- Databases are traditionally defined as repositories of facts about some aspect of the real world and a database management system (DBMS) is primarily required to simply provide an environment that is both convenient and efficient to use in retrieving and storing database information. It is implied that the DBMS should provide accurate answers to queries on the data, i.e., answers that are consistent with the real world. In fact, accurate answers are essential in nearly all common applications of databases.
- DBMS database management system
- One example of "small" error tolerance may be given in the context of market analysis performed by a large multi-national corporation to analyze sales data over the past several years to find a nation with a potential market for the company's products.
- the first step in the analysis is to compute an aggregate, such as the total dollar value of sales in order to rank the continents, and there is likely little interest in answers accurate to the last cent.
- an interesting continent is selected, a much more detailed analysis is performed on the nations within that continent, this time requiring an exact answer in order to prepare a report.
- a second example may be given in the telecommunications area, where telecommunication switches are used to route calls based on current traffic load on various available channels. Obviously, speed is of the essence in this situation. Since the traffic data can be very large, the switches typically query a dynamically maintained summary of the current load. Due to the approximate nature of the data, the switch may sometimes select a sub-optimal route, but this is not a critical hazard to the switch's operation, as long as it is not too frequent.
- Statistics such as samples and histograms, are used extensively by many DBMSs to perform critical estimations. For example, query optimizers rely on accurate selectivity estimates in order to identify an optimal execution plan and parallel query execution engines benefit from estimates of query result distributions. In these applications, statistics are used to approximate the frequency distributions of attributes in the database relations, and almost never used for query answering.
- Sampling-based statistical techniques for providing a series of increasingly accurate answers culminating in the correct answer to aggregate queries are known. Their main emphasis, however, is on efficient query processing techniques and probabilistic guarantees on the partial results of a given query, and not on using statistics to provide a single approximate answer to OLAP queries very quickly.
- Histograms are usually used to approximate the frequency distributions of one or more attributes of a relation by grouping the data into subsets (buckets) and making uniformity assumptions within each subset.
- a histogram is a bar graph in which the area of each bar is proportional to the frequency or relative frequency presented. The main advantages of histograms are that they incur almost no run-time overhead, they do not require the data to fit a probability distribution or a polynomial and, for most real-world databases, there exist histograms that produce low-error estimates while occupying reasonably small space.
- OLAP users find it useful to organize data along several dimensions of a multidimensional data cube and perform aggregate analysis on (possibly subsets of) the dimensions.
- the cells of the data cube contain the corresponding value of a measured attribute.
- the data cube is very large (several gigabytes) and answering even simple queries requires significant amounts of time. This problem has been addressed by precomputing parts of the data cube, building indexes, and using efficient techniques for computing the cube and answering queries. All of this, however, deals with providing exact, not approximate, answers to queries.
- An OPTIMAL algorithm for minimizing error given space operates as follows.
- the total number of possible histogram configurations that can be generated on N sub-cubes given a certain amount of space ⁇ is combinatorial in ⁇ ,N. This value is very high for most realistic values of ⁇ .
- the OPTIMAL algorithm identifies each possible allocation, computes its error Es (sum of Es over all sub-cubes), and finds the histogram configuration that minimizes the total error as the best. Although this algorithm is accurate, it requires huge resources, as it is clearly exponential in N.
- Another known algorithm uses a standard randomized technique, such as iterative improvement. The optimum value over all starting points in this RANDOM algorithm is then picked as the "optimal" solution.
- a third known algorithm is a NAIVE space allocation scheme, which divides the available space, ⁇ , equally among the n histograms, one on each sub-cube, Si.
- This algorithm takes a constant amount of time but it does not take the interactions between histograms on different attribute combinations into account.
- a query on Si would be answered by the direct operation on histogram H(Si, ⁇ /n).
- the resultant configurations obtained through this algorithm are not as good as other configurations because they do not take advantage of the dependencies between sub-cubes.
- the present invention alleviates to a great extent the above shortcomings in the art.
- the present invention provides a novel and unique method of summarizing database data in order to provide quick and approximate answers to aggregate queries by precomputing a summary of the data cube using histograms and answering queries using the substantially smaller summary. Further, the present invention provides a unique method identifying accurate histogram classes and distributing the space among the histograms on various sub-cubes such that the errors are minimized, while at the same time computer resources are maximized.
- FIG. 1 shows a flow chart of a method of providing approximate answers to aggregate queries according to a preferred embodiment of the present invention
- FIG. 2 illustrates three different methods of defining a histogram configuration in a database
- FIG. 3 shows the actual and approximate distribution of a data cube into data subsets (buckets);
- FIG. 4 shows an algorithm for calculation of the benefit
- FIG. 5 shows the GREEDY algorithm for bucket allocation among histograms
- FIG. 6 illustrates a database implementing the instant method of providing approximate answers to aggregate queries in a computer system.
- FIG. 1 illustrates a flow chart of a method of providing approximate answers to aggregate queries in a relational database.
- the method comprises step 9 of posing an aggregate query to a relational database.
- a description of step 10, precomputing a summary of the data cube (relational database), using sub-cubes follows below.
- Precomputing step 10 may be implemented by one of several known multi-dimensional histograms, according to efficient partitioning techniques for building them.
- Key properties that characterize histograms, including properties that determine the effectiveness of histograms in approximating aggregate distributions are the following: the sort parameter, which specifies the order in which the attribute-value/measure pairs of the data distribution are grouped in the histogram; the histogram class, which specifies the sizes of the data subsets (buckets) allowed in the histogram; the source parameter, which specifies the quantity that the histogram should try to capture accurately; and the partition constraint, which is the mathematical rule that specifies where exactly the histogram boundaries will fall based on the source parameter.
- Both the sort and the source parameters are functions of the attribute-value/measure pairs in the data distribution. Examples include the attribute value itself (V), the measure itself (M), the area (A), and the spread (S).
- the partition constraints include the following:
- Equi-sum In an equi-sum histogram with ⁇ buckets, the sum of the source values in each bucket is approximately the same and equal to
- V-Optimal In a V-Optimal histogram, the weighted sum of the variances of the source parameters values in each bucket is minimized, where the weights are equal to the number of values in the corresponding buckets.
- MaxDiff In a MaxDiff histogram, there is a bucket boundary between two source parameter values that are adjacent (in sort parameter order) if the difference between these values is one of the ⁇ -1 largest such differences.
- p(s,u) denotes a histogram class with partition constraint p, sort parameter s, and source parameter u.
- MHIST uses an MHIST algorithm for multi-dimensional histograms, although it should be obvious to one of ordinary skill in the art that any other multi-dimensional histogram algorithm that accomplishes the same or similar functions may be used in lieu of MHIST.
- MHIST repeatedly partitions the most "critical" dimension at each step (chosen based on the partition constraint from the current set of distributions), until the number of partitions equals the number of buckets needed. For example, to build a MaxDiff(V,F) histogram, this algorithm selects the dimension whose marginal distribution contains the maximum difference between any two neighbors and splits that dimension between those values.
- FIG. 3 illustrates the MaxDiff(VM) partitioning of space into five data subsets (buckets). Dashed lines 31, 32, 33, and 34 denote the order in which the splitting took place during actual distribution, while dashed lines 41, 42, 43, and 44 denote the partitioning of data during approximate distribution.
- a further step 19 in the method is the step of summarizing the different sub-cubes using histograms.
- Histograms are known to have been successfully used in approximating frequency distributions. Most aggregate measure distributions share the key properties with frequency distributions, namely, multi-dimensionality, skew in the measure (frequency) and value distributions, and dependence between attributes (defined shortly). It follows, therefore, that histograms may be used for approximating aggregate measure distributions as well.
- the exact methodology, described below, is novel and more space-efficient than the traditional usage of histograms.
- any query on a data-cube can be answered approximately by posing that query on a summary of its aggregate distribution.
- a specific class of query operations should be considered as an example.
- a common operation in OLAP queries involves applying an aggregate operator over a selected region of data in a subset of dimensions, say, S. The result of this operation can be estimated from a summary of the aggregate distribution of S.
- the Direct estimation technique 11 estimates a summary of the aggregate distribution of S by using a histogram on S.
- Slicing estimation technique 12 estimates the aggregate distribution of S by protecting the summarized distribution of a superset of S, similar to obtaining the sub-cubes from the cube, as illustrated below.
- the slicing operation enables answers to queries on many sub-cubes using a single histogram and hence reduces the number of histograms to be built.
- the accuracy of a "slice-estimate" is likely to be less than the accuracy of a "direct-estimate” because of the following reasons: (a) the sliced histogram is built on higher dimensional data which is in general more complex to approximate and (b) it may not be as accurate on each subset of attributes as the most accurate histograms built directly on those data subsets (buckets).
- Merging estimation technique 13 estimates a summary of the aggregate distribution of S from histograms on its component attributes by assuming that these attributes are independent. A formal definition of attribute independence is given below.
- a set of attributes Xi, 1 ⁇ i ⁇ n have mutually independent aggregate distributions if
- m (. . . , k, . . . , 1 . . . )/m (. . . , k, . . . , n, . . . ) m (. . . , m, . . . , 1 . . . )/m (. . . , m, . . . , n1 . . . )
- step 19 comprises defining a histogram configuration of a data cube as any set of histograms built on its sub-cubes that allows estimation on all the sub-cubes using the direct, slicing, merging and/or any other comparable technique for estimating the aggregate distribution of S.
- H(D) sub-cube D
- H(D) the histogram built on all dimensions of sub-cube D
- H(D) is characterized by its class from the taxonomy and the space allocated to it.
- Examples of valid configurations on a data cube ⁇ S,C,P ⁇ are ⁇ H(SCP) ⁇ , ⁇ H(S), H(C), H(P) ⁇ , ⁇ H(SC), H(S), H(P) ⁇ .
- ⁇ H(P), H(SP) ⁇ does not constitute a configuration because it can not be used to summarize C.
- FIG. 2 graphically illustrates the different ways of summarizing the distribution on PS.
- Line 25 from node PSC to node PS represents the slicing operation.
- Line 21 from node PS to node PS represents the direct operation.
- the merging operation is represented by line 22 from the merged node P, S (via lines 23, 24) to node PS.
- step 20 illustrates the proposition that to achieve highest accuracy on a given set of queries over a sub-cube, the summarization that minimizes the average error over these queries should be used.
- Step 20 comprises calculation of the average error given a set computer resource, i.e., space.
- the least average error on sub-cube Si(1 ⁇ i ⁇ N) by e i * is the sum of average errors in answering queries on various sub-cubes, i.e.,
- the error/space computations are made according to a novel GREEDY algorithm.
- This algorithm produces a solution close to OPTIMAL in terms of quality, and involves, as an initial step, the addition of x buckets to a histogram on a sub-cube.
- the error associated with the configuration M will be denoted by E(M).
- the benefit of allocating an additional x buckets of space to H(si) in M to generate a new configuration M', denoted as B(s i , x, M), is the reduction of error in answering queries on the data cube.
- B(s i , x, M) E(M')-E(M).
- Step 51 involves generating a first configuration, CONFIGURATION A, of a data sub-cube in a relational database using one of several known multi-dimensional histograms.
- step 52 and additional predetermined number of x buckets are allocated to CONFIGURATION A.
- step 53 involves calculating a second configuration, CONFIGURATION B for the sub-cube, using x.
- the errors for both configurations are calculated at step 54.
- Step 55 involves subtracting the calculated error values and step 56 involves updating CONFIGURATION A based on the subtracted calculated error value, thereby increasing the benefit for CONFIGURATION A.
- the GREEDY Algorithm for bucket Allocation Among Histograms shown in FIG. 5 is described as follows.
- a check of the remaining buckets is performed, at which point it is determined whether the remaining N number of buckets is larger than, or equal to, zero. If N is larger than zero, the histogram H with the maximum benefit is then identified at step 82. A new bucket is allocated to histogram H at step 83.
- the number of remaining buckets N is decreased by one.
- a check of the remaining buckets 81 is again performed. If the number of remaining buckets equals zero, the GREEDY algorithm ends and the optimal allocation of buckets among histograms has been an achieved.
- GREEDY operates, as illustrated below, in the following manner:
- the GREEDY algorithm is linear in the amount of space allocated and in the number of sub-cubes. Hence it is much more efficient than the OPTIMAL algorithm. It also takes the interactions between histograms on several sub-cubes into account.
- A(vi) denote the attribute set associated with sub-cube vi.
- A(vj) to be a function from fij, I + ⁇ R, where fij (x) is the average error in answering queries Qj on vj using a x-bucket histogram on sub-cube vi, denoted H(vi, x).
- Error functions capture the interactions between histograms under the direct and slicing operations.
- step 17 After computation of the error/space benefit using the GREEDY algorithm for each of the Direct, Slicing, and Merging techniques, the method illustrated in FIG. 1 goes on to step 17, calculating the maximum of the three error/space benefits obtained. As pointed out above, the maximum benefit corresponds to the minimum error. Thus, step 18 of the method in FIG. 1 uses the configuration having the maximum benefit of the minimum error to provide an approximate answer to the aggregate query posed at step 9.
- the computer system 60 includes a central processing unit (CPU) 71 that communicates with system 60 via an input/output (I/O) device 61 over a bus 66.
- I/O input/output
- the computer system 60 also includes random access memory (RAM) 63, read only memory (ROM) 64, and may include peripheral devices such as a floppy disk drive 70 and a compact disk (CD) ROM drive 67 which also communicate with the CPU 71 over the bus 66.
- RAM random access memory
- ROM read only memory
- CD compact disk
- the processor 71 performs logical and mathematical operations required by the method of the present invention as illustrated in FIG. 1, such as data manipulation and comparisons, as well as other arithmetic and logical functions generally understood by those of ordinary skill in the art.
- the RAM 63 is used to store the aggregate query 9, the particular histograms used at each step and program instructions required to implement the method of the present invention as illustrated in FIG. 1, and can be comprised of conventional random access memory (RAM), bulk storage memory, or a combination of both, as generally understood by those of ordinary skill in the art.
- the I/O device 61 is responsible for interfacing with an operator of the computer system 60 or with peripheral data devices (not shown) to receive or output data as generally understood by those of ordinary skill in the art.
- FIG. 1 of the present invention can reside as a computer program on a computer readable storage medium, such as a floppy disk 69 or CD ROM 67, which communicates with the CPU 71 as generally understood by those skilled in the art.
- novel method of the present invention is also extensible to several other classes of statistics (e.g., sampling), resources (e.g., usage time), and applications (e.g., selectivity estimation).
- statistics e.g., sampling
- resources e.g., usage time
- applications e.g., selectivity estimation
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Abstract
Description
∀1≦i,j≦n, ∀1≦k,m≦Di, ∀1≦l,n≦Dj,
m.sub.1, . . . ,n =1/T.sup.n-1 xm.sub.1 x . . . m.sub.n.
Es=Σe.sub.j *
B+Es=EU
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